Karmakar Smarajit, Lerner Edan, Procaccia Itamar
Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel.
Phys Rev E Stat Nonlin Soft Matter Phys. 2010 Aug;82(2 Pt 2):026105. doi: 10.1103/PhysRevE.82.026105. Epub 2010 Aug 10.
We derive expressions for the lowest nonlinear elastic constants of amorphous solids in athermal conditions (up to third order), in terms of the interaction potential between the constituent particles. The effect of these constants cannot be disregarded when amorphous solids undergo instabilities such as plastic flow or fracture in the athermal limit; in such situations the elastic response increases enormously, bringing the system much beyond the linear regime. We demonstrate that the existing theory of thermal nonlinear elastic constants converges to our expressions in the limit of zero temperature. We motivate the calculation by discussing two examples in which these nonlinear elastic constants play a crucial role in the context of elastoplasticity of amorphous solids. The first example is the plasticity-induced memory that is typical to amorphous solids (giving rise to the Bauschinger effect). The second example is how to predict the next plastic event from knowledge of the nonlinear elastic constants. Using the results of our calculations we derive a simple differential equation for the lowest eigenvalue of the Hessian matrix in the external strain near mechanical instabilities; this equation predicts how the eigenvalue vanishes at the mechanical instability and the value of the strain where the mechanical instability takes place.
我们根据组成粒子之间的相互作用势，推导出非晶态固体在无热条件下（直至三阶）最低非线性弹性常数的表达式。当非晶态固体在无热极限下经历诸如塑性流动或断裂等不稳定性时，这些常数的影响不可忽视；在这种情况下，弹性响应会大幅增加，使系统远远超出线性范围。我们证明，现有的热非线性弹性常数理论在零温度极限下收敛于我们的表达式。我们通过讨论两个例子来推动计算，在这两个例子中，这些非线性弹性常数在非晶态固体的弹塑性背景下起着关键作用。第一个例子是无定形固体典型的塑性诱导记忆（产生包辛格效应）。第二个例子是如何根据非线性弹性常数的知识预测下一个塑性事件。利用我们的计算结果，我们推导出了外部应变接近力学不稳定性时海森矩阵最低本征值的一个简单微分方程；该方程预测了本征值在力学不稳定性处如何消失以及力学不稳定性发生时的应变值。
